In this paper we develop two topics
and show their inter- and cross-relation. The first centers on
general notions of the generalized classical signal theory on
finite Abelian hypergroups. The second concerns the generalized
quantum hyperharmonic analysis of quantum signals (Hermitean
operators associated with classical signals). We study classical
and quantum generalized convolution hypergroup algebras of
classical and quantum signals.
We study classical and quantum harmonic analysis of phase space
functions (classical observes) on finite Heisenberg group HW2N+1(ZNmn, ZNmn, Zmn) over the ring Zmn. This group is the discrete version of the real Heisenberg group HW2N+1(RN, RN, R), where R is the real field. These functions have one dimensional and m1, m2,...,mn-dimensions matrix valued spectral components (for irreducible representations of HW2N+1. The family of all 1D representations gives classical world world (CW). Various mi-dimension representations (i=1,2,...,n) map classical world (CW) into quantum worlds QW}mi of ith resolution i=1,2,...,n). Worlds QW(m1) and QW(mn) contain rough information and fine details about quantum word, respectively. In this case the Fourier transform on the Heisenberg group can be considered as Weyl quantization multiresolution procedure. We call this transform the natural quantum Fourier transform.